instantly decodable network codes for real-time applications
DESCRIPTION
Instantly Decodable Network Codes for Real-Time Applications. Presented by Marios Gkatzianas. Anh Le, Arash Tehrani , Alex Dimakis , Athina Markopoulou UC Irvine, USC, UT Austin. Real-Time Applications with Wireless Broadcast. Real-time applications that use wireless broadcast: - PowerPoint PPT PresentationTRANSCRIPT
Instantly Decodable Network Codes for Real-Time Applications
Anh Le, Arash Tehrani, Alex Dimakis, Athina Markopoulou
UC Irvine, USC, UT Austin
Presented by Marios Gkatzianas
Real-time applications that use wireless broadcast:• Live video streaming• Multiplayer games
Unique characteristics:• Strict deadlines• Loss tolerant
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Problem: Retransmissions in the presence of loss
Real-Time Applications with Wireless Broadcast
Broadcast Loss Recovery for Real-Time Applications
Strict deadlines:• Instantly decodable
network codes (IDNC)
Loss tolerant:• We formulate Real-Time IDNC
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What is a coded packet that is instantly decodable and innovative to the maximum number of users?
Related Work
• Instantly decodable, opportunistic codes [Katti ’08] [Keller ’08] [Sadeghi ’09] [Athanasiadou ’12]
• IDNC focuses on minimizing the completion delay[Sorour ’10, 11, 12]
• Index coding and data exchange problems[Birk ’06] [El Rouayheb ’10]
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Our Contributions
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We show Real-Time IDNC is NP-Hard• Equivalent to finding a Max Clique in an IDNC graph• Provide a reduction from Exact Cover by 3-Sets
Analysis of instances with random loss probabilities• Provide a polynomial time solution for
Random Max Clique and Random Real-Time IDNC
Outline
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1. Real-Time IDNC• Equivalence to Max Clique• NP-Hardness
2. Random Real-Time IDNC• Optimal coded packet (clique number)• Coding algorithm
Real-Time IDNC: Problem Formulation
- A set of m packets, broadcast by a source- n users, interested in all packets- Each user received only a subset of packets
- To recover loss:
What is a coded packet that is instantly decodable and innovative to the maximum number of users?
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Real-Time IDNC: Example- 6 packets: p1 , … , p6
- 3 users: u1 , u2 , u3
p1 p2 p3 p4 p5 p6
• u1 has p1 , p2 0 0 1 1 1 1
• u2 has p3 , p5 1 1 0 1 0 1
• u3 has p3 , p6 1 1 0 1 1 0
p5 + p6 is instantly decodable and innovative to u2 and u3
p2 + p3 is instantly decodable and innovative to all users
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Mapping: Real-Time IDNC to Max Clique
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Real-Time IDNC ≡ Max Clique in IDNC graph
Real-Time IDNC is NP-Hard
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Real-Time IDNC ≡ Max Clique ≡ IQPNP-Hard
IQP is NP-Hard: reduction from Exact Cover by 3-Sets
Real-Time IDNC ≡ Integer Quadratic Programming (IQP)
Outline
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1. Real-Time IDNC• Equivalence to Max Clique• NP-Hardness
2. Random Real-Time IDNC• Optimal coded packet (clique number)• Coding algorithm
Random Real-Time IDNC
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Setup: iid loss probability p• aij = 1 with probability p
Analysis sketch:• Fix a set of j columns• Define a good row as having one 1 among these j columns
o A row is good with probability f(j) = j p (1-p) j-1
o Number of good rows has Binomial distribution: Bin(n, f(j))
Analysis of Random Real-Time IDNC
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Lemma 5 (sketch):The size of the maximum clique that touches j columns is close to the number of good rows w.r.t. these j columns
Lemma 6 (sketch):The size of the maximum clique that touches j columns concentrates around n f(j)
Theorem 7 (sketch):
Maximum clique, that touches any j columns, has size concentrating around n f(j)
Analysis of Random Real-Time IDNC (cont.)
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Corollary 8:The maximum clique touches j* columns, where j* = argmax f(j)
Observations: • j* is a constant for a fixed loss rate p • j* increases as loss rate p decreases (more packets should be coded together)
Max-Clique Algorithm
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Observations: • The maximum clique concentrates around n f(j*)• j* is a constant for a fixed loss rate p
Polynomial-time algorithm to find the maximum clique and optimal coded packet:• Examine all cliques that touch j columns
for all j δ-close to j*• Complexity: O (n m j* + δ )
Evaluation Results
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Simulation with random loss rate (20 users, 20 packets)
Max-Clique outperforms all other algorithms at any loss rate
Conclusion
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We formulate Real-Time IDNC• Equivalent to Max Clique in IDNC graph• NP-Hard proof
Analysis of Random Real-Time IDNC• Polynomial time solution to find
max clique and optimal coded packet